Bochner-Almost Periodicity for Stochastic Processes

Abstract

We compare several notions of almost periodicity for continuous processes defined on the time interval I = R or I = [0, +∞) with values in a separable Banach space E (or more generally a separable completely regular topological space): almost periodicity in distribution, in probability, in quadratic mean, almost sure almost periodicity, almost equi-almost periodicity. In the deterministic case, all these notions reduce to Bochner-almost periodicity, which is equivalent to Bohr-almost periodicity when I = R, and to asymptotic Bohr-almost periodicity when I = [0, +∞).

Almost automorphy and various extensions for stochastic processes

“…The converse implications are false. Indeed, Example 3.1 shows that a process which is almost automorphic in distribution is not necessarily almost automorphic in probability or in p-mean, see also [3,Counterexample 2.16] 3 .…”

Almost periodicity and periodicity for nonautonomous random dynamical systems

“…Furthermore, if the topology of X is defined by a family (d i ) i∈I of semidistances, we can define almost periodicity using these semidistances. The following result (see, e.g., [6,Lemma 4.4]) will also be useful in the sequel. Proposition 3.5 (almost periodicity for a family of semidistances).…”

“…In the case of random processes, each of these notions forks into several possible notions, mainly: in distribution (in various senses), in probability (or in p-mean), or in path distribution. Surveys on such notions in the case of Bohr almost periodicity can be found in [6,35].…”

Almost periodic and periodic solutions of differential equations driven by the fractional Brownian motion with statistical application

“…The application of almost periodicity to stochastic differential equations in the framework of Itô calculus seems to start in the 1980s with the Romanian school, in a series of papers by Constantin Tudor and his collaborators: [10,13,22,27,28], to cite but a few. Each known notion of almost periodicity for deterministic functions forks into several possible definitions for stochastic processes, mainly: almost periodicity in distribution (in various senses), in probability, or in square mean, see the surveys by Tudor [29] and Bedouhene et al [7]. However, almost periodicity in probability or in square mean appeared to be inapplicable to stochastic differential equations, see [4,20].…”